II. Hierarchical Evolutive Systems (HES)

Transitivity condition: a component$C ∈ H$ is a maximal set of objects linked by transitions.

Figure - a Hierarchical Evolutive System (HES) $K$

3. Multiplicity Principle

Multiplicity Principle: for each time and each level, there is at least one multi-faceted component, i.e. two patterns with the same colimit that are not isomorphic in the category of patterns and clusters.

Emergence of complex links

Complexification

“Standard changes” of living systems (cf. René Thom): birth: (addition of new components), death (addition of certain components), collision (formation of new pattern bindings), scission (destruction of certain bindings).

Emergence Theorem

Complexification preserves the Multiplicity Principle.

In a HES, there may be:

formation of multi-faceted components more and more complex

⟶ emergence of complex links between them.

III. Memory Evolutive Neural Systems

Flexible Memory

vs. Hopfield networks

Hopfield rule:

to store patterns $\textbf{p}_1, ⋯, \textbf{p}_n$ in the Hopfield network, the weight matrix is set to:

$$W = \frac 1 N \sum\limits_{ i } \textbf{p}_i \textbf{p}_i^\T$$

cf. Jupyter Notebook for simulations

Hopfield networks:

the stored patterns are not apparent in the network

pattern capacity: $≃ 0.138 N$ (where $N$ is the number of neurons)

no “flexibility”: if your mom’s appearance changes when she ages (which is unfortunately likely to happen), you don’t recognize her anymore!

spurious patterns: linear combinations of odd number of patterns also stored!